Acceleration due to gravity: Difference between revisions

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An object with mass ''m'' near the surface of the Earth experiences a downward gravitational
Where ''g'' is the '''acceleration due to gravity''', an object with mass ''m'' near the surface of Earth experiences a downward gravitational force of magnitude ''mg''. The quantity ''g'' has the dimension of acceleration, m s<sup>&minus;2</sup>, hence its name. Equivalently, it can be expressed in terms of force per unit mass, or N/kg in SI units.
force of magnitude ''mg'', where ''g'' is the '''acceleration due to gravity'''. The quantity ''g'' has the dimension of acceleration, m
s<sup>&minus;2</sup>, hence its name.


[[Gravitation#Newton's law of universal gravitation|Newtons's gravitational law]] gives the following formula for ''g'',
[[Gravitation#Newton's law of universal gravitation|Newton's gravitational law]] gives the following formula for ''g'',
:<math>
:<math>g = G\, \frac{M_{\mathrm{E}}}{R^2_{\mathrm{E}}},</math>
  g = G \frac{M_{\mathrm{E}}}{R^2_{\mathrm{E}}},
where ''G'' is the universal gravitational constant,<ref> Source: [http://physics.nist.gov/cgi-bin/cuu/Value?bg|search_for=Gravitational  CODATA 2006, retrieved 2/24/08 from NIST website]</ref> ''G'' = 6.67428 &times; 10<sup>&minus;11</sup>
</math>
 
where ''G'' is the universal gravitational constant, ''G'' = 6.67428
&times; 10<sup>&minus;11</sup>
m<sup>3</sup> kg<sup>&minus;1</sup> s<sup>&minus;2</sup>,
m<sup>3</sup> kg<sup>&minus;1</sup> s<sup>&minus;2</sup>,
''M''<sub>E</sub> is the total mass of the Earth, and ''R''<sub>E</sub>
''M''<sub>E</sub> is the total mass of Earth, and ''R''<sub>E</sub> is the radius of Earth. This equation gives a good approximation, but is not exact. Deviations are caused by the [[centrifugal force]] due to the rotation of Earth around its axis, non-sphericity of Earth, and the non-homogeneity of the composition of Earth. These effects cause ''g'' to vary roughly &plusmn; 0.02 around the value 9.8 m s<sup>&minus;2</sup> from place to place on the surface of Earth. The quantity ''g'' is therefore referred to as the ''local gravitational acceleration''. It is measured as 9.78 m s<sup>&minus;2</sup> at the equator and 9.83 m s<sup>&minus;2</sup> at the poles.
is the radius of the Earth. This equation gives a good approximation,
but is not exact. Deviations are caused by the [[centrifugal force]]
due to the rotation of the Earth around its axis, non-sphericity of the
Earth, and the non-homogeneity of the composition of the Earth. These
effects are the cause that ''g'' varies roughly &plusmn;0.01 around the
value 9.8 m s<sup>&minus;2</sup> from place to place on the surface of the Earth.
The quantity ''g'' is therefore referred to as the ''local gravitational
acceleration''.
 
The 3rd General Conference on Weights and Measures (Conférence Générale
des Poids et Mesures, CGPM) defined in 1901 a standard value denoted as
''g<sub>n</sub>''.<ref>[http://physics.nist.gov/Document/sp330.pdf The
International System of Units (SI), NIST Special Publication 330, 2001
Edition] (pdf page 29 of 77 pdf pages)
</ref><ref>[http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf#page=51
Bureau International des Poids et Mesures] (pdf page 51 of 88 pdf
pages)</ref> The value of the ''standard acceleration due to gravity'' ''g<sub>n</sub>''
is 9.80656 m s<sup>&minus;2</sup>.
 
 
 
<!--
In the [[science]]s, the term '''acceleration due to gravity''' refers  to a quantity '''g''' describing the strength of the local gravitational field. The quantity has dimension of [[acceleration]], i.e., m/s<sup>2</sup> (length per time squared) whence its name.
 
In  the article on [[Gravitation#Gravitational potential|gravitation]] it is shown that for a relatively small altitude ''h'' above the surface of a large, homogeneous, massive sphere (such as a planet) [[Isaac Newton|Newton's]] [[Gravitation#Gravitational potential|gravitational potential]] ''V'' is to a good approximation linear in ''h'':  ''V''(''h'') = ''g h'', where '''g''' is the '''acceleration due to gravity'''. This aproximation relies on ''h'' << ''R''<sub>sphere</sub> (where ''R''<sub>sphere</sub> is the radius of the sphere). The exact gravitational potential is not linear, but is inversely proportional to the distance, ''r'', from the centre of the Earth: <math>V_G = \frac{G M}{r}</math>. 
 
On Earth, the term ''standard acceleration due to gravity'' refers to the value of 9.80656 m/s<sup>2</sup> and is denoted as '''g<sub>n</sub>'''. That value was agreed upon by the 3rd General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM) in 1901.<ref>[http://physics.nist.gov/Document/sp330.pdf The International System of Units (SI), NIST Special Publication 330, 2001 Edition] (pdf page 29 of 77 pdf pages) </ref><ref>[http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf#page=51 Bureau International des Poids et Mesures] (pdf page 51 of 88 pdf pages)</ref> The actual value of acceleration due to gravity varies somewhat over the surface of the Earth;  '''g''' is referred to as the ''local gravitational acceleration'' .
 
Any object of [[mass]] ''m'' near the [[Earth]] (for which the [[altitude]] ''h'' << ''R''<sub>Earth</sub>) is subject to a [[force]] ''m g'' in the downward direction that causes an [[acceleration]] of magnitude '''g<sub>n</sub>''' toward the surface of the earth.  This value serves as an excellent approximation for the local acceleration due to [[gravitation]] at the surface of the earth, although it is not exact and the actual acceleration '''g''' varies slightly between different locations around the world.
-->
<!--
More generally, the acceleration due to gravity refers to the magnitude of the force on some test object due to the mass of another object. Under [[Gravitation#Newton's law of universal gravitation|Newtonian gravity]] the gravitational field strength,  due to a [[spherical symmetry|spherically symmetric]] object of mass ''M'' is given by:
:<math>f = G \frac{M}{r^2}. </math>
The magnitude of the acceleration ''f'' is expressed in [[SI]] units of [[meter]]s per [[second]] squared. Here ''G'' is the [[universal gravitational constant]] ''G'' = 6.67428&times;10<sup>&minus;11</sup> Nm<sup>2</sup>/kg<sup>2</sup> <ref> Source: [http://physics.nist.gov/cgi-bin/cuu/Value?bg|search_for=Gravitational  CODATA 2006, retrieved 2/24/08 from NIST website]</ref> and <math>r</math> is the distance from the test object to the centre of mass of the Earth and ''M'' is the mass of the Earth.  
 
In [[physics]], it is common to see [[acceleration]] as a vector, with an absolute value (magnitude, length) ''f''  and a direction from the test object toward the center of mass of the Earth (antiparallel to the position vector of the test object), hence as a vector the acceleration is:


:<math>
The 3rd [[General Conference on Weights and Measures]] (Conférence Générale des Poids et Mesures, CGPM) defined in 1901 a standard value denoted as ''g<sub>n</sub>''.<ref>[http://physics.nist.gov/Document/sp330.pdf The International System of Units (SI), NIST Special Publication 330, 2001 Edition] (pdf page 29 of 77 pdf pages)</ref>
\vec{f} = - G \frac{M}{r^2} \vec{e}_r \quad \hbox{with}\quad \vec{e}_r \equiv \frac{\vec{r}}{r}.  
<ref>[http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf Bureau International des Poids et Mesures] (Brochure on SI, pdf page 51 of 88 pdf pages) From the website of the [[Bureau International des Poids et Mesures]]</ref> The value of the ''standard acceleration due to gravity'' ''g<sub>n</sub>'' is 9.80665 m s<sup>&minus;2</sup>. This value of ''g<sub>n</sub>'' was the conventional reference for calculating the now obsolete unit of force, the kilogram force, as the force needed for one kilogram of ''mass'' to accelerate at this value.
</math>
-->


==References==
==References==
<references/>
{{reflist}}


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Where g is the acceleration due to gravity, an object with mass m near the surface of Earth experiences a downward gravitational force of magnitude mg. The quantity g has the dimension of acceleration, m s−2, hence its name. Equivalently, it can be expressed in terms of force per unit mass, or N/kg in SI units.

Newton's gravitational law gives the following formula for g,

where G is the universal gravitational constant,[1] G = 6.67428 × 10−11 m3 kg−1 s−2, ME is the total mass of Earth, and RE is the radius of Earth. This equation gives a good approximation, but is not exact. Deviations are caused by the centrifugal force due to the rotation of Earth around its axis, non-sphericity of Earth, and the non-homogeneity of the composition of Earth. These effects cause g to vary roughly ± 0.02 around the value 9.8 m s−2 from place to place on the surface of Earth. The quantity g is therefore referred to as the local gravitational acceleration. It is measured as 9.78 m s−2 at the equator and 9.83 m s−2 at the poles.

The 3rd General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM) defined in 1901 a standard value denoted as gn.[2] [3] The value of the standard acceleration due to gravity gn is 9.80665 m s−2. This value of gn was the conventional reference for calculating the now obsolete unit of force, the kilogram force, as the force needed for one kilogram of mass to accelerate at this value.

References

  1. ↑ Source: CODATA 2006, retrieved 2/24/08 from NIST website
  2. ↑ The International System of Units (SI), NIST Special Publication 330, 2001 Edition (pdf page 29 of 77 pdf pages)
  3. ↑ Bureau International des Poids et Mesures (Brochure on SI, pdf page 51 of 88 pdf pages) From the website of the Bureau International des Poids et Mesures
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